Baire category theorem

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The Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space.

The theorem was proved by René-Louis Baire in his 1899 doctoral thesis.[1]

Contents

Statement of the theorem

A Baire space is a topological space with the following property: for each countable collection of open dense sets Un, their intersection ∩ Un is dense.

Note that neither of these statements implies the other, since there is a complete metric space which is not locally compact (the irrational numbers with the metric defined below), and there is a locally compact Hausdorff space which is not metrizable (uncountable Fort space). See Steen and Seebach in the references below.

  • (BCT3) A non-empty complete metric space is NOT the countable union of nowhere-dense sets (i.e., sets whose closure has dense complement).

This formulation is a consequence of BCT1 and is sometimes more useful in applications. Also: if a non-empty complete metric space is the countable union of closed sets, then one of these closed sets has non empty interior.

Relation to the axiom of choice

The proofs of BCT1 and BCT2 for arbitrary complete metric spaces require some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to a weak form of the axiom of choice called the axiom of dependent choices.[2]

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